Hostname: page-component-74656f9884-c52kr Total loading time: 0 Render date: 2026-11-09T01:55:29.606Z Has data issue: false hasContentIssue false
  • English
  • Français

A bivariate counting process

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Ray Watson  and
Paul Yip
Ray Watson*
Affiliation:
University of Melbourne
Paul Yip*
Affiliation:
University of Hong Kong
*
∗ Postal address: Department of Statistics, University of Melbourne, Parkville, VIC 3052, Australia.
∗∗ Postal address: Department of Statistics, University of Hong Kong, Pokfulam Road, Hong Kong.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a bivariate Markov counting process with transition probabilities having a particular structure, which includes a number of useful population processes. Using a suitable random time-scale transformation, we derive some probability statements about the process and some asymptotic results. These asymptotic results are also derived using martingale methods. Further, it is shown that these methods and results can be used for inference on the rate parameters for the process. The general epidemic model and the square law conflict model are used as illustrative examples.

Keywords

POPULATION PROCESSRANDOM TIME-SCALE TRANSFORMATIONMARTINGALEINFERENCE

MSC classification

Primary: 60J99: None of the above, but in this section

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 2 , June 1993 , pp. 353 - 364
Copyright
Copyright © Applied Probability Trust 1993 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Aalen, O. O. (1978) Non-parametric inference for a family of counting processes. Ann. Statist. 6, 701–726.Google Scholar
Andersen, P. K. and Borgan, Ø. (1985) Counting process models for life history data: A review. Scand. J. Statist. 12, 97–158.Google Scholar
Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases. Griffin, London.Google Scholar
Gye, R. and Lewis, T. (1976) Lanchester's equations: mathematics and the art of war. A historical survey and some new results. Math. Scientist. 1, 107–119.Google Scholar
Watson, R. (1981) A useful random time scale transformation for the standard epidemic model. J. Appl. Prob. 17, 324–332.Google Scholar
Whittle, P. (1955) The outcome of a stochastic epidemic – a note on Bailey's paper. Biometrika 42, 116–122.Google Scholar
Yip, P. and Watson, R. (1991) An inference procedure for conflict models. Stoch. Proc. Appl. 37, 161–171.Google Scholar